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Theorem bj-nnst 13624
Description: Double negation of stability of a formula. Intuitionistic logic refutes unstability (but does not prove stability) of any formula. This theorem can also be proved in classical refutability calculus (see https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See nnnotnotr 13872 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in  (  ->  ,  -.  ) -intuitionistic calculus with definitions (uses of ax-ia1 105, ax-ia2 106, ax-ia3 107 are via sylibr 133, necessary for definition unpackaging), and in  (  ->  ,  <->  ,  -.  )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)
Assertion
Ref Expression
bj-nnst  |-  -.  -. STAB  ph

Proof of Theorem bj-nnst
StepHypRef Expression
1 bj-trst 13620 . 2  |-  ( ph  -> STAB  ph )
2 bj-fast 13622 . 2  |-  ( -. 
ph  -> STAB  ph )
31, 2bj-imnimnn 13619 1  |-  -.  -. STAB  ph
Colors of variables: wff set class
Syntax hints:   -. wn 3  STAB wstab 820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605
This theorem depends on definitions:  df-bi 116  df-stab 821
This theorem is referenced by:  bj-stst  13626  bj-dcst  13642
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